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A dynamical system is ergodic if it preserves a measure and each measurable invariant set is a zero set or the complement of a zero set. No measurable invariant set has intermediate measure. See also Section 6. The classic real world example of ergodicity is how gas particles mix. At time zero, chambers of oxygen and nitrogen are separated by a wall. When the wall is removed the gasses mix thoroughly as time tends to in¯nity. In contrast think of the rotation of a sphere. All points move
along latitudes, and ergodicity fails due to existence of invariant equatorial bands. Ergodicity is stable if it persists under perturbation of the dynamical system. In this paper we ask: "how common are ergodicity and stable ergodicity?" and we propose an answer along the lines of the Boltzmann hypothesis "very." There are two competing forces that govern ergodicity hyperbolicity and the KAM phenomenon. The former promotes ergodicity and the latter impedes it. One of the striking applications of KAM theory and its more recent variants is the existence of open sets of volume preserving dynamical systems each of which possesses a positive measure set of invariant tori and hence fails to be ergodic. Stable ergodicity fails dramatically for these systems. But does the lack of ergodicity persist if the system is weakly coupled to another? That is, what happens if you have a KAM system or one of its perturbations that refuses to be ergodic, due to these positive measure sets of invariant tori, but somewhere in the universe there is a hyperbolic or partially hyperbolic system weakly coupled to it? does the lack of egrodicity persist? The answer is "no," at least under reasonable conditions on the hyperbolic factor.

By: Charles Pugh, Michael Shub

Published in: American Mathematical Society. Bulletin, volume 41, (no 1), pages 1-41 in 2004


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